Characteristic-Based Deep Learning Framework for Hamilton-Jacobi Equations and Optimal Transport

Learn about a novel deep learning methodology for solving Hamilton-Jacobi partial differential equations through this 36-minute conference presentation by Stanley Osher from UCLA. Discover how to derive an implicit solution formula for Hamilton-Jacobi PDEs using the method of characteristics, closely related to classical Hopf-type formulations. Explore the development of a deep learning framework that computes viscosity solutions without requiring supervised training data, leveraging neural networks' mesh-free nature and expressive capacity for scalable solutions to high-dimensional and nonconvex problems. Understand how this approach enables efficient optimal transport model construction by exploiting the characteristic structure of Hamilton-Jacobi equations, where bidirectional optimal transport maps can be represented in closed form through associated HJ equation solutions. Examine the direct computation of optimal transport maps using the implicit solution formula, eliminating numerical ODE integration requirements and achieving substantial improvements in transport map accuracy and sampling process efficiency.